Charles University in PragueFaculty of Mathematics and Physics

ABSTRACTof Doctoral Thesis

Mgr. Pavel Stransky

Classical and Quantum Chaosin Atomic Nuclei

Institute of Particle and Nuclear Physics

Supervisor: Doc. RNDr. Pavel Cejnar, Dr.Branch: F10 - Nuclear Physics

Univerzita Karlova v PrazeMatematicko-fyzikalnı fakulta

AUTOREFERAT

doktorske prace

Mgr. Pavel Stransky

Klasicky a kvantovy chaosv atomovych jadrech

Ustav casticove a jaderne fyziky

Skolitel: Doc. RNDr. Pavel Cejnar, Dr.Studijnı program: F10 - jaderna fyzika

Vysledky tvorıcı doktorskou praci byly zıskany behem doktorskeho studia na Mate-maticko-fyzikalnı fakulte Unverzity Karlovy v Praze v letech 2004–2009.

Doktorand: Mgr. Pavel Stransky

Ustav casticove a jaderne fyzikyV Holesovickach 2180 00 Praha 8

Skolitel: Doc. RNDr. Pavel Cejnar, Dr.

Ustav casticove a jaderne fyzikyV Holesovickach 2180 00 Praha 8

Skolıcı pracoviste: Ustav casticove a jaderne fyzikyV Holesovickach 2180 00 Praha 8

Oponenti: Doc. Ing. Frantisek Becvar, DrSc.Katedra fyziky nızkych teplotV Holesovickach 2180 00 Praha 8

Doc. Ing. Zdenek Pluhar, CSc.

Ustav casticove a jaderne fyzikyV Holesovickach 2180 00 Praha 8

Tento autoreferat byl rozeslan dne: ............

Obhajoba doktorske prace se kona dne ............ v ............ hodin pred Komisı proobhajoby doktorskych pracı v oboru F10 – Jaderna fyzika na MFF UK, V Holesovickach2, Praha 8 v mıstnosti c. ............

S doktorskou pracı je mozne se seznamit na studijnım oddelenı dekanatu MFF UK(oddelenı pro doktorske studium), Ke Karlovu 3, Praha 2.

Predseda RDSO F10: Prof. RNDr. Jan Kvasil, DrSc.

Ustav casticove a jaderne fyzikyV Holesovickach 2180 00 Praha 8

4

Contents

1 Introduction 6

2 Theoretical outline 8

2.1 Geometric collective model . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Scaling properties and phase diagram . . . . . . . . . . . . . . . . 9

2.1.3 Quantization and diagonalization . . . . . . . . . . . . . . . . . . 10

2.2 Measuring and visualising chaos . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Fraction of regularity . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Spectral statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Peres method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Numerical results 14

3.1 Classical Chaos in the GCM . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Quantum chaos in the GCM . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Brody parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.2 Peres lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Conclusion 22

References 24

List of original work 26

5

Chapter 1

Introduction

Although models of collective dynamics of atomic nuclei [1, 2] belong to the oldestsubject of nuclear physics, they still exhibit enormous potential for yielding importantnew results. Many of them is connected with a modern form of the collective model,the interacting boson model (IBM) [3], but purely geometric approach still providesadequate framework for the interpretation of main features of nuclear collective motions,even those accommodated within the IBM.

In the present work, we use the geometric collective model (GCM) to study theinterplay between regular and chaotic behavior both in the classical and quantum case.The definition of chaos is primarily derived from the instability of classical motions,expressing the organization of trajectories in the phase space. The term “quantumchaos” [4] refers to some quantum properties—most commonly specific short- and long-range correlations in spectra of energy eigenvalues—that usually accompany the classicalchaoticity. Intense study of these properties in the last decades made it possible to extendthe notion of chaos from macroscopic systems to small quantum objects like atoms ornuclei.

Historically, nuclear physics offered the first confrontation of ideas related to quan-tum chaos with experimental data. It was when sequences of neutron resonances witha fixed spin and parity Jπ were shown to agree with spectral correlations predicted bythe random-matrix theory [5]. The persistence of the same kind of statistics was sub-sequently demonstrated [6] also for ensembles of low-energy nuclear levels (although inthis case also serious deviations from chaotic correlations were detected in some familiesof states, particularly the collective states with Jπ = 2+ and 4+ in even-even nuclei).Possible existence of chaotic layers of dynamics was recently discussed [7] even in con-nection with ground states of nuclei—in the analysis of fluctuating properties of nuclearmasses.

All these data suggest that atomic nuclei at low energies exhibit a variable mixtureof regular and chaotic dynamical aspects. However, very little is known about the prin-cipal mechanisms that control the competition between both types of motions. Severalpossible sources of irregularity can be identified already on the mean-field level, butthe most substantial effects are supposed to result from residual interactions, which,however, are too difficult for explicit analysis.

6

A viable alternative to the attempts to solve the nuclear many-body problem in itsfull complexity is the use of simplified models that capture only some essential dynamicalfeatures of nuclei in various regimes [1]. Since residual interactions are responsible forhighly correlated collective modes of motions (complementary to single-particle modes),one may try to search for signatures of regularity and chaos in the known elementarycollective models, such as the IBM or GCM. This approach was pioneered by Alhassidet al. [9, 10] and by Paar et al. [11] using the IBM.

A detailed study of ordered and chaotic motions in the framework of the geometriccollective model is presented in this work. The thesis is based particularly on publishedreferences [I, II, VII, VIII].

This abstract begins with some theoretical remarks in Chapter 2 which exposesbasics of the geometric collective model and techniques of measuring chaos. We restrictourselves to a subset of quantum levels with zero angular momentum, which makes theconfiguration space effectively two-dimensional, in contrast to five-dimensional spacecorresponding to general motions. Then we proceed to the classical version of the model(Sec. 3.1) and describe the dependence of classical measure of regularity on energy andparameters of the model and continue towards the quantum version in Sec. 3.2.

The model for zero rotations can be quantized in two physically meaningful ways.We solve the eigenvalue problem in both cases and compare the level statistics obtained,looking particularly into the regions where transitions between regular and chaotic dy-namics take place. The possibility of changing the value of a classicality (Planck) con-stant enables us to populate the spectrum with variable density of quantum states, whichis used for a global inspection of large energy domains and zooming in some finer de-tails. By comparing the classical and quantum measures of regularity and the quantummeasure calculated in various quantization schemes we test the Bohigas’ conjecture [12].

We also employ the more or less forgotten graphical method invented in 1984 byPeres [13], which turns out to be very fruitful as an indicator of the changing structuresacross the spectrum containing possibly a very large number of states. The spectrumof stationary states of a given quantum system with two degrees of freedom is drawnas a lattice in the plane E × 〈P 〉, where E is energy and 〈P 〉 stands for an arbitraryobservable average. This allows one to recognize ordered and disordered patterns andvisually allocate regular and chaotic domains within the same energy interval. Finally,the abstract is closed by the conclusion.

7

Chapter 2

Theoretical outline

2.1 Geometric collective model

2.1.1 Hamiltonian

We consider the Hamiltonian of the geometric collective model (GCM) [14] in the trun-cated form

H =

√5

2K[π × π](0) +

+√

5 A [α × α](0) −√

35

2B

[

[α × α](2) × α

](0)

+ 5 C(

[α × α](0))2

(1)

where αµ, µ = −2, . . . , 2 form the spherical tensor of quadrupole shape variables (theycan be viewed as coefficients at the Y2µ spherical harmonics in an expansion of nuclearradius), having five independent real components, πµ = −K(dα∗

µ/dt) the associated

tensor of momenta, and [•× •](λ) stands for the coupling to angular momentum λ. Theconstants {K,A,B,C} are adjustable parameters. We neglect higher-order terms of theHamiltonian, in particular the 3rd-order kinetic term.

The transformation into the principal frame of deformation yields the potential inthe form

V = Aβ2 + Bβ3 cos 3γ + Cβ4 =

= A(

x2 + y2)

+ B(

x3 − 3xy2)

+ C(

x2 + y2)2

, (2)

where we use the usual parametrization [1]:

α0 ≡ x = β cos γ√

2 α2 ≡ y = β sin γ. (3)

The system described by Eq. (1) has five degrees of freedom. The tensor α can bediagonalized to get two shape coordinates x and y (or β and γ) and three Euler angles{θ1, θ2, θ3} characterizing the relative orientation of the principal and laboratory frames.

8

Parameters β and γ represent the known Bohr variables; β = 0 stands for sphericalshape, β > 0 implies nonzero deformation with prolate or oblate axially deformed shapefor γ equal to even or odd multiples of π/3, respectively.

We restrict ourselves to the the nonrotating case J = 0: the principal frame is atrest and the motion can be described solely in terms of x and y (or equivalently in termsof β and γ). In this case our system is effectively two dimensional (2D).

It is worth mentioning that the system is integrable for B = 0 and the parameter Bgoverns the strength of nonintegrable perturbation. We denote the nonintegrable termof the potential (2) as

H ′ = β3 cos 3γ. (4)

2.1.2 Scaling properties and phase diagram

Hamiltonian (1) contains four external parameters. Without the loss of generality, thefundamental physical units of the system can be chosen such that three out of the fourparameters become equal to unity or to ±1 for A, while the remaining parameter takesthe role of the external variable. The only fundamental parameter, determining the typeof equilibrium shape of the nuclei, can be introduced as

τ ≡ AC

B2. (5)

In the quantum case, the classicality parameter

κ ≡ ~2

K, (6)

whose changes can be viewed either as changes of the Planck constant, or as changes ofthe mass, constitutes the second independent parameter of the model. This parameterdetermines the absolute density of states.

In the whole work we fix C = 1 and K = 1.

Figure 1 shows the phase structure of the geometric model. Gray parabolae τ =const. (or paraboloids, if the third coordinate of the parametric space correspondingto the parameter C is considered) represent sets of parameters giving scale-equivalentHamiltonians. Thick solid lines A < 0, B = 0 and τ = 1/4 depict first-order phasetransitions between prolate and oblate shape and between deformed and spherical shape,respectively. Dotted line, which can be expressed explicitly as

A = −1, B ∈ (0, 1)

A ∈ (−1, 1), B = 1 (7)

A = 1, B ∈ (1, 0),

represents the curve which crosses all parabolae. It is profitable to present the resultson this path.

9

� 1.0 � 0.5 0.5 1.0

� 2

� 1

1

2

-1

-1

-2 -3 -4 -5 5 4 3

1

2

-2 -3 -4 -5 5 4 3 2

1

B

A

-

- -

-

Figure 1: The GCM phase diagram in the plane of parameters A and B for any C > 0. Thicksolid lines represent phase separatrices (first-order), thick dashed lines enclose the region ofphase coexistence. Thin gray parabolae correspond to dynamically equivalent parameter setsrelated by scaling transformations. They are marked with values ±1/

√

|τ |. Dotted line depictsthe path which passes through all different configurations of the GCM system.

2.1.3 Quantization and diagonalization

Standard quantization procedure with πµ = −i~ ∂∂αµ

(satisfying comutation relation

[αµ, πν ] = i~δµν) leads to the kinetic term [1]

T (5D) = − ~2

2K

(

1

β4

∂

∂ββ4 ∂

∂β+

1

β2 sin 3γ

∂

∂γsin 3γ

∂

∂γ

)

+ Trot, (8)

where Trot stands for a nontrivial rotational part of the kinetic energy, containing thederivatives with respect to Euler angles and coupling all five dynamical variables (this isthe reason for the notation 5D). However, in the considered nonrotating case this termis identically zero, Trot = 0, and only the two vibrational degrees of freedom remain.

For J = 0 the truly 2D scheme represents an alternative way of quantization. Itmeans that we put the system into the intrinsic frame first and only then carry out thequantization. The kinetic term obtained in this way reads as follows:

T (2D) = − ~2

2K

(

1

β

∂

∂ββ

∂

∂β+

1

β2

∂2

∂γ2

)

, (9)

which is nothing but the standard 2D kinetic energy expressed in polar coordinates(r, φ) ≡ (β, γ).

We diagonalize both types of the GCM Hamiltonian in the respective 2D or 5Dharmonic oscillator bases. While the original 5D solution possesses several implicitsymmetries arising from the ambiguity of the system’s orientation in the intrinsic frame,solutions of the 2D model do not a priori satisfy such symmetries. If the spectra and

10

spectral statistics associated with both quantizations are to be compared, conditions ofsymmetries need to be imposed externally also to the 2D case. In addition, if we slightlyrelax the conditions, we can split the 2D quantization scheme into the odd and evencases, which gives us another independent class of 2D solutions.

Both expressions (8) and (9) have the same classical limit and offer a possibility tostudy the influence of the quantization method on the quantum chaotic features of thespectrum. Let us stress that in the nuclear physics context only the 5D quantization iscorrect.

2.2 Measuring and visualising chaos

2.2.1 Fraction of regularity

At first we introduce the measure of classical regularity. Trajectories in the phase spaceof an integrable system with 2 degrees of freedom lie on surfaces that are topologicallyequivalent to 4D tori. In crossings with a plane (the so called Poincare section) such toriform sets of 1D curves (ovals) which are covered by passages of individual trajectories(periodic orbits create just sequences of isolated points on these curves). On the contrary,a fully chaotic system is subject to the ergodic principle which ensures that orbitsuniformly cover the whole kinematically accessible area of the Poincare section. Inintermediate cases—for partially chaotic/regular systems—the randomly covered areasin the Poincare section coexist with regular islands, remnants of tori. An example of aPoincare section is shown in Fig. 2.

-1 0 1

-1

0

1

px

x

Figure 2: An example of the Poincare section. Each dot corresponds to the passage of onetrajectory of the GCM system with parameters A = −1, B = 0.446, and energy E = 0 throughthe plane y = 0 (in this figure we show altogether 50000 such crossings of 52 randomly chosentrajectories).

11

We quantify the degree of regularity in case of J = 0 by a relative fraction of regularislands in the Poincare section. More specifically, we introduce the ratio

freg =Sreg

Stot

, (10)

where Sreg is the area corresponding to all regular islands in the y = 0 Poincare sectionand Stot is the total kinematically accessible area of the same section. Fraction (10) maychange between limiting values 0 (complete chaos) and 1 (perfect order) as the externalcontrol parameter A and energy E vary.

Regular fraction freg would be determined through dividing the total area of the sec-tion into an infinite grid of cells and labeling each cell according to whether it is crossedby a regular or chaotic trajectory, which are distinguished by the Smaller alignmentindices method (SALI) [15]. More details concerning the calculation of freg are given inarticles [I, II, V].

2.2.2 Spectral statistics

In this subsection we turn to measuring quantum chaos. According to Bohigas’ con-jecture [12], chaotic systems exhibit strong correlations between levels, which result inan effect known as “spectral rigidity”. The short-range component of these correlationsis most clearly manifested in the distribution of the nearest-neighbor spacings (NNS),i.e. gaps between neighboring levels in a transformed (so-called unfolded) spectrum. Infully chaotic systems, this distribution is amazingly well approximated by the Wignerdistribution, while in systems with regular classical counterparts the NNS distributiontends to be Poissonian.

A suitable quantity allowing one to interpolate between the two limiting cases is theBrody parameter ω [16]. It is defined through the distribution

PB(s; ω) = (ω + 1)Nω sω exp(

−Nω sω+1)

(11)

Nω =

[

Γ

(

ω + 2

ω + 1

)]ω+1

,

where s is the spacing between adjacent levels in the unfolded spectrum and Nω afactor resulting from the required conditions

∫ ∞

0PB(s; ω)ds = 1 (normalization) and

∫ ∞

0sPB(s; ω)ds = 1 (unfolding). Eq. (11) interpolates between the Poisson (ω = 0)

and Wigner (ω = 1) distributions, hence a value ω ∈ 〈0, 1〉 obtained from a concretespectrum tells us where between order and chaos the actual system is. The adjunct(1−ω) of the Brody parameter can be directly compared with the regular fraction freg.

Practically we proceed as follows: We divide the calculated spectra (usually we areable to find 30000 well-converging levels) into windows of 1000 levels and after unfoldingprocedure we estimate the Brody parameter for each window via the χ2 fit, applied tothe cumulative distribution function

RB(s; ω) ≡∫ s

0

PB(s′; ω)ds′ = 1 − e−Nω sω+1

. (12)

As a result of this procedure we obtain energetical dependence of the Brody parameter.

12

2.2.3 Peres method

Quantum measures of chaos are usually based on certain statistical properties of thespectra of energy levels, e.g. on the nearest-neighbor-spacing distribution (and therelated Brody parameter ω, see previous section) or on the long range correlations ofeigenvalues [17]. It is clear that the statistics describes only some bulk features of asufficiently large portion of the spectrum. However, the method introduced by Peres in1984 [13] makes it possible to assign regular, chaotic, or a mixed type of dynamics toindividual states. Peres’ method is entirely visual and can be considered as a quantumanalogue of the classical technique based on the Poincare sections.

The method can be briefly described as follows: Having an arbitrary constant ofmotion A in an integrable conservative system and drawing its eigenvalues Ai againstenergies Ei of individual levels i, the resulting image forms a fully regular lattice (dueto the EBK quantization). Since the constant of motion A can be unknown or evendoes not exist (in nonintegrable systems), Peres proposed to take the time average ofan arbitrary observable P . The operator corresponding to this average commutes withthe Hamiltonian, or its classical Poisson brackets vanishes. In practice it means that wecalculate expectation values 〈P 〉i for individual eigenstates and plot them against Ei. Itcan be shown that in an integrable (regular) system such a lattice is formed by orderedpatterns, while chaotic (or mixed) systems exhibit (partly) disordered lattices.

In the geometric model we consider two different Peres operators. The first one isidentified with the square of the angular momentum operator L connected with therotations varying angle γ. In the 2D case, this is the Casimir invariant of the O(2)algebra of rotations in the (β, γ) plane:

L22D = ~

2 ∂2

∂γ2. (13)

In the 5D case, L2 is the Casimir invariant of the GCM algebra O(5) [18] restricted tovalue J = 0 of the O(3) angular momentum (null rotations in the ordinary space). Wehave

L25D =

~2

sin 3γ

∂

∂γsin 3γ

∂

∂γ. (14)

The second Peres operator used in our analysis correspond to the nonintegrable pertur-bation H ′ of the Hamiltonian, given by Eq. (4).

13

Chapter 3

Numerical results

3.1 Classical Chaos in the GCM

The fraction of regularity freg for given parameter and energy is calculated by means ofthe method described in Sec. 2.2.1. The complete map of freg in the nonrotating caseof the GCM system (1) is depicted in Fig. 3. The horizontal axis shows the values of afree parameter of the system and is divided by thick white lines into three regions withdifferent scalings, following the path expressed by Eq. (7). The vertical axis shows theenergy. The regularity with extremely interesting behavior at absolute energy E = 0,which corresponds for A < 0 to a local maximum of the potential (2) at β = 0, ispresented separately in Fig. 4.

Both leftmost and rightmost parts of the figure correspond to the integrable caseB = 0 with freg = 1. As no explicit second integral of motions is known for B 6= 0,we expect increasing chaos when the integrable limit B = 0 is being departed. This is,however, not quite so. The dependence of freg on energy and a control parameter of thesystem exhibits very complex nonmonotonous behavior, involving virtually integrableregions. The most important well-pronounced island of regularity in a wide range ofenergies is observed at B ≈ 0.6. As shown in Ref. [III], this region is connected withthe so-called regular arc of the interacting boson model [9, 10]. Increased values of theregular fraction are also observed for other values of the adjustable parameter at E ≈ 0,see Fig. 4.

The degree of chaos also depends on energy. For E close to the minimal value ofthe potential (2), the system performs nearly harmonic vibrations around one of theminima. For E ≫ 0, one again expects regular behavior as the quartic term in potential(2) completely prevails (see Refs. [I, II]). In the intermediate region of energies, strikingareas with high regularity is observed. The bands of full regularity—the strong oneclearly visible at energy E ≈ 24 for A ' −1.2 in the scaling B = 1 and followed by asmaller one at E ≈ 7 for A ' 0.6 in the same scaling—still remain unexplained.

Dash black line in the Fig. 3 encircles the area where the kinematically accessibleregion have concave border. In the geometric model, the lower part of the line (upto energies E ≈ 1) corresponds exactly with the curve obtained by the method usingRiemannian geometry [19], which should separate regions with regular and chaotic dy-

14

10

00

10

0

10 1

0.1

0.0

1

0.0

01

10

00

10

0

10

1 0.1

0.0

1

0.0

01

00

.51 -1

-0.5

00

.511

AEE B

0

-0.1

-0.2

-0.3

-0.4 E

0 -0.1

-0.2

-0.3

-0.4 E

f re

g

00

.20

.40

.80

.61

0.5

0

A =

-1

B =

1A

= 1

Figure 3: Complete map of classical regular fraction freg as a function of control parametersand energy E. The degree of chaos is coded in shades of gray, with light gray (black) corre-sponding to complete order (chaos). The plot is divided by the thick white lines into threeregions with different scaling (A = −1, B = 1, and A = 1, respectively), according to thepath expressed by Eq. (7). White dashed line indicates the deformed-spherical shape phasetransition. Black dashed line corresponds to the convex-concave transition of the border ofkinematically accessible area (concave shape is confined inside). The absolute error of thecalculation of freg does not exceed 0.02.

15

0.0

0.5

1.0

1 0 0.5-1 -0.5 0

BA

freg

Figure 4: The regular fraction freg for E = 0 as a function of B (scaling A = −1, the left partof the graph) and of A (scaling B = 1, the right part of the graph).

namics. However, it is evident from the figure that this geometrical criterion gives onlyapproximate estimation of the place where regularity falls into chaos. Generally oneobserves that chaotic regions slightly penetrate out through the separation line withsome exceptions appearing, e.g., near the arc of regularity B ≈ 0.6.

3.2 Quantum chaos in the GCM

3.2.1 Brody parameter

We compare the quantum measure of regularity (Brody parameter), obtained in the waydescribed in Sec. 2.2.2, with the corresponding classical measure freg from Fig. 3.

Figure 5 depicts the dependence of both measures freg and (1 − ω) on energy. Thevalue of the Hamiltonian parameter B was chosen as B = 1.09 The energy range shownin the figures represents the most interesting region, lying between the domains of fullregularity at very low and very high energies. As was mentioned above, for energiesjust above the global minimum of the potential the system is entirely regular due tothe validity of the harmonic-well approximation. With increasing energy, the regularitysuddenly breaks down and continues falling sharply until it nearly reaches zero. Afterthis stage, it takes off again and exhibits several well pronounced peaks of highly regularmotions, which are separated by valleys of more chaotic dynamics. For sufficiently highenergies, not shown in the present figure, the regularity starts growing steadily towardthe fully regular limit, following roughly a logarithmic dependence.

Classical measure freg, taken from the vertical section of the Fig. 3 at B = 0.62, isdrawn by the solid line, the energy dependence of the adjunct of the Brody parameter

16

0 10 20 300.0

0.5

1.0

freg

1- (2D E) 1- (5D)

regu

larit

y

E

Figure 5: A comparison between the classical regular fraction freg and an adjunct (1 − ω) ofthe Brody parameter for B = 1.09 calculated in two different types of quantization (dash linecorrespond to the 2D even case, dotted line to the 5D case).

(1−ω) by the dashed and dotted line for 2D even and 5D quantization scheme, respec-tively. Observed maxima and minima of all curves coincide, however, it turns out thatthe Brody parameter tends to slightly overestimate the regularity. Indeed, there is noreason to expect that freg and (1− ω) behave in exactly the same way. We only expecta qualitative agreement, and that is fully confirmed in the present calculation.

In this figure we can also observe the dependence of the quantum measure of chaoson the method of quantization. Although the spectra obtained by various types ofquantizations differ from each other distinctly, the Brody parameter for all quantizationschemes exhibits essentially the same dependence on energy, the mutual deviations beingfully within the range of standard errors. Therefore, we can report that full agreementwith Bohigas’ conjecture is confirmed in the present model independently of the chosenquantization scheme.

More detailed discussion concerning the correspondence of classical and quantummeasures of chaos of is presented in the article [VII].

3.2.2 Peres lattices

We construct the Peres lattices in the GCM system for two independent operators L2

and H ′, see Sec. 2.2.3.

First of all, we show the decay of regularity with increasing perturbation B, depictedin Fig. 6. Panel (a) demonstrates the fully regular lattices for the integrable case B = 0(the average value 〈H ′〉 is identically zero in this case). It is worth emphasizing that thelattice in an integrable case is regular for any averaged operator. As can be seen in panel

17

0.0

1.0

1.0

0.0

0.0 0.3 0.6 0.0 0.3 0.6

0.0

0.5

-0.5

0.0

0.5

-0.5

⟨L ⟩ ⟨H' ⟩

(a) B = 0

(b) B = 0.005

E

0.0

0.5

0.0 0.3 0.3-0.3 -0.3 0.0

0.0

0.5

-0.5

(c) B = 0.24

2

Figure 6: Peres lattices for J = 0 eigenstates of the GCM in the 2D even quantization (~ =0.005) with 3000 levels plotted. The points represent individual eigenstates with coordinates Ei

(energy) and 〈P 〉i (expectation value of the respective Peres operator). Two Peres operators,L2 (left column) and H ′ (right column) are taken into account. Row (a) correspond to the fullyintegrable case, B = 0. Row (b) depicts the disturbance of the lattice with small nonintegrableperturbation, B = 0.05. Row (c) shows the most chaotic case, B = 0.24.

18

(b), adding a small perturbation to the integrable Hamiltonian does not instantaneouslybreak down the entire regular lattice. Instead, some localized seeds of distortion arecreated, while the rest of the lattice remains ordered in the same fashion as in theintegrable case. This scenario is in accordance with Percival’s conjecture [20] assumingthat the sets of regular and chaotic eigenstates are statistically independent in thesemiclassical limit ~ → 0, i.e. they do not interact with each other. Therefore, thepersisting regular parts of the lattice can be associated with surviving remnants ofclassical tori, while the disordered parts correspond to proliferating chaotic orbits. Asthe perturbation strength B grows, the remnants of tori are gradually disappearing anddisorder tends to increasingly plague the lattice. This is demonstrated by an almosttotally disordered lattice in panel (c), where only a few low lying states keeps the regularpattern.

The above-outlined visual method allows one to judge which parts of a mixed spec-trum (or, in optimal cases, which individual states) are regular and which are chaotic,in contrast to traditional methods of quantum chaos based on the spectral statistics.since in that case regular and chaotic (or mixed) parts of the spectrum can only bespecified by energy. In the present approach, these parts can coexist within the sameenergy interval, the additional information needed for their separation being obtainedfrom the behavior of the averages 〈P 〉i.

0.6

0.6

0.0

0.00.0 0.0-0.5 -0.50.5 0.5

-1

-10

0

E

(a) 2D even

(b) 5D

⟨L ⟩ ⟨H' ⟩2

Figure 7: Peres lattices of the GCM at A = −1, B = 0.62 with averages of L2 (left) and H ′

(right) for 2D even (row a) and 5D (row b) quantizations (~ = 0.005). In row (a), three statesdenoted by full symbols are identified in both lattices. In the inset of the top right panel theenergetical dependence of the classical regular fraction freg is shown.

19

-1 0 1 -1 0 1x

-1 0 1

1

0

-1

y

(a) (b) (c)

Figure 8: Squared wave functions for the three marked states from Fig. 7 (2D even quantiza-tion). The states (a) and (c), which correspond to the square and the diamond, respectively,are taken from the regular part of the lattice (the 1995th and 1885th level, respectively). Thestate (b), associated with the bullet, belongs to the chaotic part (the 1890th level), showingan ergodic behavior.

E E

⟨L ⟩2⟨H' ⟩

(a) h=0.01

(b) h=0.02

Figure 9: The same as in Fig. 7, but only for 2D even quantizations and different values ofPlanck constant: ~ = 0.01 (row a) and ~ = 0.02 (row b). Dashed boxes enclose the same areaof each column as is shown in Fig. 7.

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In Fig. 7 we compare Peres lattices obtained (for both Peres operators) in the 2Deven and 5D quantizations. The value B = 0.62 belongs to the island of stronglypronounced regularity close to the resonance of β and γ vibrations, see Sec. 3.1. Theregularity shows up as a large area of ordered points, which starts at the lowest negativeenergies and spreads over to positive energies, where it is joined by a rising chaotic area.Despite the spectra for different quantizations show significant differences, the form ofPeres lattices is rather similar.

In order to demonstrate the coincidence of the regular and chaotic regions in thelattices for different Peres operators, we have highlighted three of the states in the firstrow of Fig. 7, marking them by a square, a bullet, and a diamond. Probability densitiesfor the corresponding wave functions are depicted in Fig. 8. The wave functions as wellas the location of the respective points in the Peres lattice show that the square andthe diamond correspond to regular states, while the bullet represents a chaotic state. InFig. 7(a) we see that this assignment is consistent for both choices of the Peres operator.Let us note that both regular levels (a) and (c) in Fig. 8 exhibit a large increase of thewave-function magnitude in a region where a certain periodic trajectory oscillates in theclassical case [4].

Although we can choose an arbitrary operator for plotting the Peres lattice, Fig. 7indicates that some operators may be more suitable than others. For some choices, apart of the regular region in the lattice can pervade into the chaotic area and hide therebehind a disordered mesh of points. In such cases, one cannot decide whether a levelinside a chaotic region is indeed chaotic. (On the other hand, overlapping regular areasform a regular area again.) While there is no doubt that these observations demonstratelimitations of the Peres method, one can improve its resolution by employing severalincompatible Peres operators. Indeed, Fig. 7 (right) shows that for the three highlightedstates a better choice of Peres operator is P = H ′.

Figure 9 displays Peres lattices for 2D even quantization with different Planck con-stant ~. Tuning the value of ~—which is proportional to the classicality constant κ, seeEq. 6—one scales the absolute density of quantum states. Here we want to show thatthese changes do not influence the main features of the Peres lattice. The variations ofthe lattice with ~ = 0.005, 0.01, 0.02 for both Peres operators are observed in Figs. 7and 9. The structures with well pronounced minima and maxima of the dependenceclassical regular fraction freg on energy is shown in the insets. A decrease of the ~ valueincreases the density of states (the system gets closer to the classical limit) and servesas a zoom into the sea of levels: one can see finer details of the lattice but (because ofcomputational limits) a smaller fraction of the spectrum is available. For a comparison,the thick dashed box in both panels of the same column of Fig. 9 encloses a fixed regionof energy × P-average from Fig. 7. It is seen that the structures observed in the latticebecome wealthier in details as κ decreases, but the overall appearance of the relevantpart of the lattice remains the same.

Peres lattices in the geometric collective model are fully discussed in Ref. [VIII].

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Chapter 4

Conclusion

In this work we studied J = 0 vibrations in the geometric collective model, that cap-tures the most essential collective degrees of freedom in nuclei. In spite of the model’sconceptual simplicity, it exhibits enormous complexity of solutions, with a very sensi-tive dependence of the observed behaviors on external parameters and energy. Thesefeatures qualify the model for being a subject of detailed analyses of the competitionbetween regular and chaotic modes of motions on both classical and quantum levels.

First we have analyzed classical chaos which allowed us to to understand the clas-sical character of various types of motions that emerge in different regimes of nuclearcollective dynamics, in particular those involved in the competition between regular andchaotic features. We have demonstrated that the GCM exhibits an unexpectedly com-plex interplay between regular and chaotic types of motions. The important result isthe discovery of two well-pronounced quasiregular islands in intermediate domains ofthe control parameter B and energy E. One of these islands resides at low energies,E < 1, in the range approximately B ≈ 0.6, A = −1, and the other one at intermedi-ate energies, E ≈ 24, within the interval A ∈ (−1.2,∞), see Fig. 3. While the latterquasiregular region is probably too high in excitation energy to be relevant experimen-tally, the former one is undoubtedly a close relative of the IBM “arc of regularity” [9, 10].It needs to be stressed that the origin of regularity in the above domains still remainsrather unclear. On the other hand, the mechanism underlying the emergence of chaos innuclear collective motions at the lowest energies was identified with the convex–concavetransition of the shape of the kinematically accessible area in the plane of deformationparameters, which exactly correspond to the regular-chaos transition obtained by thegeometric method [19].

The framework of quantized geometric model allowed us to study the correspondencebetween classical and quantum measures of chaos. Although the dependence of chaoticmeasures on some external control parameters has been extensively studied in quantumbilliards, see e.g. Refs. [21, 22], the dependence on energy has been marginalized sofar. This is partly because billiard systems do not permit this kind of analysis—theirchaotic features are always uniform in energy. Even the studies based on “soft” potentialsystems have been so far focused mostly on the cases with a relatively simple energydependence of chaotic measures, see e.g. Refs. [23, 24]. In this sense, the present workcan be considered as complementary to the studies based on two-dimensional billiard

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systems. We have demonstrated that a tight connection between classical and quantummeasures of chaos, embodied in the well-known Bohigas’ conjecture [12], remains valideven under the condition of a strong energy dependence.

Another important conclusion of the work is the observed independence of quantumchaotic measures on the method used to quantize the system. Since the definition ofquantum chaos is based on the system’s semiclassical limit, it would be very surprisingto find the opposite—i.e. statistical properties of spectra depending on the quantization.However, the question deserves to be further tested. The present work is probably onlya first step in this direction.

Our results show that the Brody parameter, despite of being often deprecated, repre-sents a reasonably sensitive measure of chaos in quantum system. This method enabledus to study the breakdown and reoccurrence of ordered quantal spectra with runningparameter B in a much more detailed way.

Finally we have revitalized an almost 35 years old method by Peres [13] and showedits great potential in the field of quantum chaos and even beyond. Peres lattices pro-vide an excellent viewpoint to the landscape around the border between classical andquantum physics. This is so especially for systems with two degrees of freedom whoselattices can be drawn as two-dimensional diagrams, in analogy with planar Poincaresections of such systems. If applied within the domain of quantum chaos, Peres’ methodenables one to distinguish regular and chaotic behaviors on the level of individual statesor subsets of states within the same energy interval. The structural information onindividual eigenstates is represented by a single variable (the average of a suitably cho-sen observable), which allows one to use a simple visualization technique incorporatingsimultaneously a large number of states.

We showed that we can construct for one system more lattices with different Peresoperators and that some of them are more suitable than others to capture the mostessential dynamical features. The optimal choice of the Peres operator should be sub-ject to further study. Although the Peres’ method does not directly yield a calculablemeasure of quantum chaos, it represents an important indicator providing new insightsinto the origin of chaotic behavior.

The work summarizes our long-term effort to map chaotic properties of the geometriccollective model of nuclear physics.. A great advantage of the geometric model (and alsoof the related interacting boson model) is the apparent conceptual simplicity encodingstrikingly rich complexity of dynamics. Let us note that the above simplified modelscapture the main phenomenological features of nuclear collectivity which are presentlybeside a fully microscopic description. The study of disordered collective dynamicswithin these models may be considered as an attack to the problem of chaos in many-body systems from the direction perpendicular to the mean-field approach.

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References

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List of original work

[I] P. Cejnar, P. StranskyRegular and chaotic vibrations of deformed nuclei with increasing γ-rigidityPhysical Review Letters 93, 102502 (2004)

[II] P. Stransky, M. Kurian, P. CejnarClassical chaos in the geometric collective modelPhysical Review C 74, 014306 (2006)

[III] M. Macek, P. Stransky, P. Cejnar, S. Heinze, J. Jolie, J. DobesClassical and quantum properties of the semiregular arc inside the Casten trianglePhysical Review C 75, 064318 (2007)

[IV] M. Macek, P. Stransky, P. CejnarOrder and chaos in the interacting boson modelPhysics of Atomic Nuclei 70, sv. 9, 1592 (2007)

[V] P. Stransky, P. Cejnar, M. MacekOrder and chaos in the geometric collective modelPhysics of Atomic Nuclei 70, sv. 9, 1572 (2007)

[VI] P. Cejnar, P. StranskyImpact of quantum phase transitions on excited level dynamicsPhysical Review E 78, 031130 (2008)arXiv:0807.3649 [quant-ph]

[VII] P. Stransky, P. Hruska, P. CejnarQuantum chaos in the nuclear collective model: Classical-quantum correspondencePhysical Review E 79, 046202 (2009)arXiv:0902.3983 [quant-ph]

[VIII] P. Stransky, P. Hruska, P. CejnarQuantum chaos in the nuclear collective model: Peres LatticesPhysical Review E, in pressarXiv:0902.4155v1 [quant-ph]

[IX] P. Cejnar, P. Stransky, M. MacekQuantum phase transitions and nuclear structureInternational Journal of Modern Physics E, in press

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[X] M. Macek, P. Stransky, P. CejnarPeres lattices in nuclear structureInternational Journal of Modern Physics E, in pressarXiv:0810.2949 [nucl-th]

[XI] P. Cejnar, M. Macek, P. Stransky, M. KurianRegular and chaotic nuclear vibrationsin: Capture Gamma-Ray Spectroscopy and Related Topics, ed. A. Woehr, A. Apra-hamian (AIP Conference Proceedings 819, 2006), p. 487

[XII] P. Cejnar, P. StranskyQuantum phase transitions for excited statesin: Capture Gamma-Ray Spectroscopy and Related Topics, ed. A. Blazhev, J. Jolie,N. Warr, A. Zilges (AIP Conference Proceedings 1090, 2009), p. 169

[XIII] P. Stransky, M. Macek, P. Cejnar, J. DobesPeres lattices in nuclear structure and beyondin: Capture Gamma-Ray Spectroscopy and Related Topics, ed. A. Blazhev, J. Jolie,N. Warr, A. Zilges (AIP Conference Proceedings 1090, 2009), p. 174

[XIV] Interactive web presenting the main resultshttp://www-ucjf.troja.mff.cuni.cz/∼stransky/results

Comment:

List of original work is sorted chronologically by the date of publication. The articlespublished in journals are written in black font, other works in gray font.

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